quotient rule of exponents

The quotient rule says that to divide two powers of the same base, you keep the base and subtract the exponents: b^m / b^n = b^(m−n), as long as b is not zero. Dividing cancels matching factors, and subtracting counts the ones left over.

For instance 2^5 / 2^2 = (2 × 2 × 2 × 2 × 2) / (2 × 2). Two factors on the bottom cancel two on top, leaving 2 × 2 × 2 = 2^3. Subtracting 5 − 2 = 3 gives the same answer without writing out the cancellation.

When the bottom exponent is larger, subtraction can give a zero or negative exponent — for example 2^2 / 2^5 = 2^(−3), which equals 1/2^3. This is exactly why negative exponents are defined the way they are: the rule keeps working in every case. The base must be nonzero, since dividing by zero is undefined.